3.136 \(\int \tanh ^2(c+d x) (a+b \tanh ^2(c+d x)) \, dx\)

Optimal. Leaf size=36 \[ -\frac {(a+b) \tanh (c+d x)}{d}+x (a+b)-\frac {b \tanh ^3(c+d x)}{3 d} \]

[Out]

(a+b)*x-(a+b)*tanh(d*x+c)/d-1/3*b*tanh(d*x+c)^3/d

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Rubi [A]  time = 0.04, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3631, 3473, 8} \[ -\frac {(a+b) \tanh (c+d x)}{d}+x (a+b)-\frac {b \tanh ^3(c+d x)}{3 d} \]

Antiderivative was successfully verified.

[In]

Int[Tanh[c + d*x]^2*(a + b*Tanh[c + d*x]^2),x]

[Out]

(a + b)*x - ((a + b)*Tanh[c + d*x])/d - (b*Tanh[c + d*x]^3)/(3*d)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 3473

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(b*Tan[c + d*x])^(n - 1))/(d*(n - 1)), x] - Dis
t[b^2, Int[(b*Tan[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]

Rule 3631

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp
[(C*(a + b*Tan[e + f*x])^(m + 1))/(b*f*(m + 1)), x] + Dist[A - C, Int[(a + b*Tan[e + f*x])^m, x], x] /; FreeQ[
{a, b, e, f, A, C, m}, x] && NeQ[A*b^2 + a^2*C, 0] &&  !LeQ[m, -1]

Rubi steps

\begin {align*} \int \tanh ^2(c+d x) \left (a+b \tanh ^2(c+d x)\right ) \, dx &=-\frac {b \tanh ^3(c+d x)}{3 d}-(-a-b) \int \tanh ^2(c+d x) \, dx\\ &=-\frac {(a+b) \tanh (c+d x)}{d}-\frac {b \tanh ^3(c+d x)}{3 d}-(-a-b) \int 1 \, dx\\ &=(a+b) x-\frac {(a+b) \tanh (c+d x)}{d}-\frac {b \tanh ^3(c+d x)}{3 d}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 65, normalized size = 1.81 \[ \frac {a \tanh ^{-1}(\tanh (c+d x))}{d}-\frac {a \tanh (c+d x)}{d}+\frac {b \tanh ^{-1}(\tanh (c+d x))}{d}-\frac {b \tanh ^3(c+d x)}{3 d}-\frac {b \tanh (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[Tanh[c + d*x]^2*(a + b*Tanh[c + d*x]^2),x]

[Out]

(a*ArcTanh[Tanh[c + d*x]])/d + (b*ArcTanh[Tanh[c + d*x]])/d - (a*Tanh[c + d*x])/d - (b*Tanh[c + d*x])/d - (b*T
anh[c + d*x]^3)/(3*d)

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fricas [B]  time = 0.41, size = 160, normalized size = 4.44 \[ \frac {{\left (3 \, {\left (a + b\right )} d x + 3 \, a + 4 \, b\right )} \cosh \left (d x + c\right )^{3} + 3 \, {\left (3 \, {\left (a + b\right )} d x + 3 \, a + 4 \, b\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} - {\left (3 \, a + 4 \, b\right )} \sinh \left (d x + c\right )^{3} + 3 \, {\left (3 \, {\left (a + b\right )} d x + 3 \, a + 4 \, b\right )} \cosh \left (d x + c\right ) - 3 \, {\left ({\left (3 \, a + 4 \, b\right )} \cosh \left (d x + c\right )^{2} + a\right )} \sinh \left (d x + c\right )}{3 \, {\left (d \cosh \left (d x + c\right )^{3} + 3 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + 3 \, d \cosh \left (d x + c\right )\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(d*x+c)^2*(a+b*tanh(d*x+c)^2),x, algorithm="fricas")

[Out]

1/3*((3*(a + b)*d*x + 3*a + 4*b)*cosh(d*x + c)^3 + 3*(3*(a + b)*d*x + 3*a + 4*b)*cosh(d*x + c)*sinh(d*x + c)^2
- (3*a + 4*b)*sinh(d*x + c)^3 + 3*(3*(a + b)*d*x + 3*a + 4*b)*cosh(d*x + c) - 3*((3*a + 4*b)*cosh(d*x + c)^2
+ a)*sinh(d*x + c))/(d*cosh(d*x + c)^3 + 3*d*cosh(d*x + c)*sinh(d*x + c)^2 + 3*d*cosh(d*x + c))

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giac [B]  time = 0.17, size = 86, normalized size = 2.39 \[ \frac {3 \, {\left (d x + c\right )} {\left (a + b\right )} + \frac {2 \, {\left (3 \, a e^{\left (4 \, d x + 4 \, c\right )} + 6 \, b e^{\left (4 \, d x + 4 \, c\right )} + 6 \, a e^{\left (2 \, d x + 2 \, c\right )} + 6 \, b e^{\left (2 \, d x + 2 \, c\right )} + 3 \, a + 4 \, b\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{3}}}{3 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(d*x+c)^2*(a+b*tanh(d*x+c)^2),x, algorithm="giac")

[Out]

1/3*(3*(d*x + c)*(a + b) + 2*(3*a*e^(4*d*x + 4*c) + 6*b*e^(4*d*x + 4*c) + 6*a*e^(2*d*x + 2*c) + 6*b*e^(2*d*x +
2*c) + 3*a + 4*b)/(e^(2*d*x + 2*c) + 1)^3)/d

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maple [B]  time = 0.02, size = 100, normalized size = 2.78 \[ -\frac {b \left (\tanh ^{3}\left (d x +c \right )\right )}{3 d}-\frac {a \tanh \left (d x +c \right )}{d}-\frac {b \tanh \left (d x +c \right )}{d}-\frac {\ln \left (\tanh \left (d x +c \right )-1\right ) a}{2 d}-\frac {\ln \left (\tanh \left (d x +c \right )-1\right ) b}{2 d}+\frac {\ln \left (1+\tanh \left (d x +c \right )\right ) a}{2 d}+\frac {\ln \left (1+\tanh \left (d x +c \right )\right ) b}{2 d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tanh(d*x+c)^2*(a+b*tanh(d*x+c)^2),x)

[Out]

-1/3*b*tanh(d*x+c)^3/d-a*tanh(d*x+c)/d-b*tanh(d*x+c)/d-1/2/d*ln(tanh(d*x+c)-1)*a-1/2/d*ln(tanh(d*x+c)-1)*b+1/2
/d*ln(1+tanh(d*x+c))*a+1/2/d*ln(1+tanh(d*x+c))*b

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maxima [B]  time = 0.33, size = 105, normalized size = 2.92 \[ \frac {1}{3} \, b {\left (3 \, x + \frac {3 \, c}{d} - \frac {4 \, {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} + 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + 2\right )}}{d {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} + 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} + 1\right )}}\right )} + a {\left (x + \frac {c}{d} - \frac {2}{d {\left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(d*x+c)^2*(a+b*tanh(d*x+c)^2),x, algorithm="maxima")

[Out]

1/3*b*(3*x + 3*c/d - 4*(3*e^(-2*d*x - 2*c) + 3*e^(-4*d*x - 4*c) + 2)/(d*(3*e^(-2*d*x - 2*c) + 3*e^(-4*d*x - 4*
c) + e^(-6*d*x - 6*c) + 1))) + a*(x + c/d - 2/(d*(e^(-2*d*x - 2*c) + 1)))

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mupad [B]  time = 1.17, size = 34, normalized size = 0.94 \[ x\,\left (a+b\right )-\frac {b\,{\mathrm {tanh}\left (c+d\,x\right )}^3}{3\,d}-\frac {\mathrm {tanh}\left (c+d\,x\right )\,\left (a+b\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tanh(c + d*x)^2*(a + b*tanh(c + d*x)^2),x)

[Out]

x*(a + b) - (b*tanh(c + d*x)^3)/(3*d) - (tanh(c + d*x)*(a + b))/d

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sympy [A]  time = 0.29, size = 54, normalized size = 1.50 \[ \begin {cases} a x - \frac {a \tanh {\left (c + d x \right )}}{d} + b x - \frac {b \tanh ^{3}{\left (c + d x \right )}}{3 d} - \frac {b \tanh {\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (a + b \tanh ^{2}{\relax (c )}\right ) \tanh ^{2}{\relax (c )} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(d*x+c)**2*(a+b*tanh(d*x+c)**2),x)

[Out]

Piecewise((a*x - a*tanh(c + d*x)/d + b*x - b*tanh(c + d*x)**3/(3*d) - b*tanh(c + d*x)/d, Ne(d, 0)), (x*(a + b*
tanh(c)**2)*tanh(c)**2, True))

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